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8/8/2019 I-1 DSP Review_2003
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Lecture #1Review of DT Signal
& System Concepts
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Notational Conventions (P-1.2)
Sets of Numbers: R = real numbersC = complex numbersZ = integers ( , -3, -2, -1, 0, 1, 2, 3, )
Signals: x(t ), y(t ) = continuous-time (CT) signals x[n], y[n] = discrete-time (DT) signals
Modulo Operator Clock Math: n mod N
n: -4 -3 -2 -1 0 1 2 3 4 5 6 7 n mod N: 2 0 1 2 0 1 2 0 1 2 0 1 15 oclock = 15 mod 12= 3 oclock
DT Convolution:
==
m
mn ym xn y x ][][]}[*{
DT Circular Convolution: 10]mod)[(][]}[{1
0
=
= N n N mn ym xn y x
N
m
Recall: We dont want Circular convolution it is what we get if we try to implementConvolution in the frequency domain by multiplying DFTs (but not DTFTs)But we can fix it so that it works. We use proper zero-padding!!!
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Transforms (P-2)Fourier Transform for CT Signals
{ }
=
=
=
d e X t x
dt et x
Fx X
t j F
t j
F
)(21
)(
)(
)()( { }
=
=
=
=
d e X n x
en x
Fx X
n j
n
n j
)(21
][
][
)()(
f
f DiscreteT imeFourierT ransform
Discrete Fourier Transform for DT Signals
{ }
1,...,2,1,0][1
][
1,...,2,1,0][
][][
1
0
/ 2d
1
0
/ 2
d
==
==
=
=
=
N nek X N
n x
N k en x
k Dxk X
N
k
N kn j
N
n
N kn j
Discrete Fourier Transform
Z Transform for DT Signals
{ }
=
=
=
n
n z n x
z Zx z X
][
)()(z
Set z = e j
Fourier Transform for DT
Signals
Inverse ZT done using partialfractions & a ZT table
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Fourier Transforms for DT Signals (P-2.7,P-4)DTFT = In Head/On Paper tool for analysis & design
DFT = In Computer tool for Implementation/Analysis&Design(In both cases you generally strive to make the DFT behave like DTFT)
DFT for Implementation:! Compute DFT of signal to be processed and manipulate DFT values! May or may not need zero-padding (depends on application)
DFT for Analysis & Design:!
Compute DFT of given filters TF to see its frequency response! Generally use zero-padding to get fine grid to get approximate DTFT
DFT/DTFT Properties you Must Know: Periodicity. (DFT&DTFT both periodic in FD)
. (IDFT gives periodic TD signal) Time Shift.. (Time Shift in TD " Linear Phase Shift in FD) Frequency Shift. (Mult. by complex sine in TD " Freq Shift in FD) Convolution Theorem (Conv. in Time Domain " Mult. in Freq Domain)
Multiplication Theorem.. (Mult. in Time Domain " Conv. in Freq Domain) Parsevals Theorem. (Sum of Squares in TD = Sum of Squares in FD)
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Summation Rules (P-1.3)Make sure you are familiar with rules for dealing withsummations, as shown in Section 1.3 of Porat.
In addition, a particular summation formula that shows up
often is the Geometric Summation :
=
=
= 1 if ,
1 if ,1
12
1
21
2
1
N N
N N
N
N n
n
Special Case: N 1 = 0
=
=
= 1 if ,
1 if ,1
1
2
1
0
2
2
N
N
N
n
n
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Eulers Formulas
j
ee
ee
j j
j j
2
)sin(
2)cos(
=
+=
Im
je
je
)cos(2
je 2 j s i n
( )
)sin()cos(
)sin()cos(
je
je j
j
=+=
Im
Re
je
je
)cos(
)sin( j
)sin( j Re
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Discrete-Time Processing (P-2, P-7)h[n]
H f () H z( z )
x[n] X f () X z( z )
y[n] = h[n]* x[n]Y f () = H f () X f ()Y z( z ) = H z( z ) X z( z )
h[n] = systems Impulse ResponseIt is response of system to input of [n]
H f () = systems Frequency ResponseIt is DTFT of impulse response
H z( z ) = systems Transfer FunctionIt is ZT of impulse response
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Discrete-Time System Relationships
Time Domain Z / Freq Domain
DTFT
ZT
ZT (Theory)
Inspect (Practice)
Inspect Inspect Roots
Unit Circle
z = e j
p p
qq z
z a z a
z b z bb z H
+++
+++=
L
L1
1
110
1)(
==
+=q
ii
p
ii in xbin yak y
01
][][][
Pole/ZeroDiagramBlock
Diagram
Difference
Equation
Transfer
Function
FrequencyResponseImpulse
Response je z
z z H H == |)()(f
h[k ]
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Poles and Zeros of Transfer Function
Consider Here Only p>q
)())((
)())((
)(
)1(
)(
1)(
21
21
11
110
11
110
11
110
p
qq p
p p p
qqqq p
p p
p
qq p
p p
qq z
z z z
z z z z
a z a z
b z b z b z
z a z a z
z b z bb z z
z a z a
z b z bb z H
=
+++
+++=
++++++
=
++++++
=
LL
L
L
L
L
L
L
Re{ z }
Im{ z }
Numerator Roots define zerosDenominator Roots define poles
Poles must be insideunit circle for stability
Poles & zeros must occur inComplex Conjugate pairs
to give real-valued TF coefficients
In this case pq zeros at z =0
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IIR and FIR Filters
p p
qq z
z a z a z b z bb z H
++++++=
LL
11
110
1)(
FIR : Finite Impulse Response h[n] has only finite many nonzero valuesFilter is FIR : If a i = 0 for i = 1, 2, , p (It is IIR otherwise)
z z b z bb z H +++= L110)(
==otherwise
qnbnh n,0
,...,1,0,][ TF coefficients are theimpulse response values!!
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Example System
Time Domain Z / Freq Domain
DifferenceEquation
TransferFunction
ZT (Theory)Inspect (Practice)
)(1)( 1 z X z z Y = )()1()( n xn yn y =Input-Output Form
1
1)(
)()(
==
z z X
z Y z H
)1()()( += n yn xn y
Recursion Form
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Example System (cont.)
Time Domain Z / Freq Domain
Difference
Equation
Block Diagram
ZT (Theory)
Inspect (Practice)
Inspect Inspect
11)( =
z z H
z-1
x(n) y(n)
y(n-1)
)1()()( += n yn xn y
Recursion Form
Transfer
Function
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Example System (cont.)
Z / Freq Domain
11)( =
z z H
Transfer
Function
==
j
j
e z e z
1
OnUnitCircle
Unit Circle = je z 1
],(1
)(
= je
H FrequencyResponse
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Example System (cont.)
[ ]
[ ]
[ ] [ ]
=
+=
+=
=
=
)cos(1)sin(
tan)(
)sin()cos(1)(
)sin()cos(1
)sin()cos(11)(
1
22
H
H
j
je H j
EulersEquation
Plotting Transfer Function
Group intoReal & Imag
StandardEqs for
Mag. & Angle
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Example System (cont.)
TransferFunction
Pole/ZeroDiagram
Roots
Z/Freq Domain
=
= z
z z
z H 11)(
UnitCircle
Zero
Num. = 0When z = 0
Re{z}
Im{z}
If < 1: In side UCIf > 1: Out side UC
Pole
Den. = 0When z =
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Example System (cont.)
ImpulseResponse
DTFT
ZTUnit Circle
11)( =
z z H
],(
1)(
= je H
n
nh =)(
Inv. ZT
Table(See Porat)
=
d enh j1)(
Inv. DTFT
Time Domain Z / Freq Domain
TransferFunction
FrequencyResponse
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Review of StandardSampling Theory
(P-3.1-3.2)
P ti l S li S t U
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0 0 .2 0 . 4 0 .6 0 . 8 1- 2
- 1
0
1
2
Time
S i g n a
l V a
l u e
0 0 .2 0 . 4 0 .6 0 . 8 1- 2
- 1
0
1
2
Time
S i g n a
l V a
l u e
PulseGen
CT LPF
==
n pnT t pnT xt x )(~)()(~
)(~ t xr
DAC
HoldSample at
t = nT
x(t )
ADC
T = Sampling Interval F s = 1/ T = Sampling Rate
Practical Sampling Set-Up
x[n] = x(nT )
Sampling Analysis (#1)
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Sampling Analysis (#1)
Goal = Determine Under What Conditions We Have:
Reconstructed CT Signal = Original CT Signal )()(~
t xt x =
)()(~ t t p =Simplify to Develop Theory: Use
Why???? 1. Because delta functions are EASY to analyze!!!2. Because it leads to the best possible case
=
=n
p p nT t nT xt xt x )()()()(~
x p(t ) is called the Impulse Sampled signal
Note: x p(t ) shows up during Reconstruction, not Sampling!!
Sampling Analysis (#2)
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Sampling Analysis (#2)Now, the impulse sampled signal x p(t ) is filtered to givethe reconstructed signal xr (t )
CT LPF
=
=
=
=
n
n p
nT t t x
nT t nT xt x
)()(
)()()(
)()()(
???)(
f X f H f X
t x
pr
r
=
=b
)( f H
Need to find X p( f ). But First a Trick!!!
=
=
=
=
=
=
k
t kF j
k
T kt j
n p
set xT
eT
t x
nT t t xt x
2
/ 2
FSuseperiodic
)(1
1)(
)()()(
4 4 34 4 21 { }
=
=
=
=
k
s
k
t kF j p
kF f X T
et x F T
f X s
)(1
)(1
)( 2
Sampling Analysis (#3)
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Sampling Analysis (#3)
ImpulseGen CT LPF x[n] = x(nT ))(t x p )(t xr
DAC
HoldSample at
t = nT
ADC
f
f
f
X ( f )
X p( f )
X r ( f )
F s 2F s F s 2 F s
B B
X r ( f ) = X ( f )If F s 2 B
x(t )
H ( f )
f
Sampling Analysis (#4)
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Sampling Analysis (#4)
What this says: Samples of a bandlimited signal completely define
it as long as they are taken at F s 2 BImpact: To extract the info from a bandlimited signal we only need tooperate on its (properly taken) samples
# Use computer to process signals
Hold
Sample att = nT
x(t ) x[n] = x(nT ) Extracted
InformationComputer
Sampling Analysis (#5)
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Sampling Analysis (#5)
FT of Impulse Sampled Signal gives view of Original FT
&FT of Impulse Sampled Signal = DTFT of Samples
#DTFT gives view of FT of original signal
Lets see this
T en xenT x
nT t F nT xnT t nT x F t x F
n
jn
n
T jn
n en p
T jn
===
==
=
=
=
=
][)(
)}({)()()()}({ 4 4 34 4 21
)() / ( f F p X T X =
Sampling Analysis (#6)
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Sampling Analysis (#6)
f F
T
s=
=
2
X ( f )
DTFT
B B f
f F s F s /2 2F s F s 2 F s F s /2
DT Freq(rad/sample)2 4 2 4
/ Normalized DT Freq2 4 2 4 11
Read notes on web on Concept of Digital Frequency
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Introduction toEE521 Case Study
d
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EE521 Case Study: Emitter Location
Data Link
DataLink
Receiver #3(Rx3)
Processing Tasks Intercept RF Signal @ Rxs Sample Signal Suitably for Processing Detect Presence of Emitters Signal
Estimate Characteristics of Signal Use Estd Chars to Classify Emitter Share Data Between Rxs Cross-Correlate Signals to Locate Tx
Radio Freq Transmitter (Tx) Communications Radar
Receiver #1(Rx1) Receiver #2
(Rx2)
Rx1Overall Processing Set-Up
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RFFront-End
DigitalFront-End
CrossCorrelation
DetectSignal
Estimate
SignalParameters
ClassifyEmitter
Data Link &Decompress
Antenna
Rx2RF Data Link
SamplingSub-System
DigitalAnalog
RFFront-End
DigitalFront-End
DetectSignal
Estimate
SignalParameters
ClassifyEmitter
Antenna
Compress& Data Link
Rx1
SamplingSub-System
DigitalAnalog
Processing Tasks Overview (#1)
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RF Front-End (Analog Circuitry): RF = Radio Frequency Selects reception band; may need to scan bands
Amplifies signal to level suitable for ADC, etc. Frequency-shifts signal spectrum to range suitable for ADC
! Technology trend is toward requiring less shifting Unfortunately: noise introduced by analog electronics ( Random Signals )
Topic Well Cover
RF
Front-End200 MHz
FT of Signal
Before RF-FE
3 GHz1 GHz 4 GHz
Antenna
FT of SignalAfter RF-FE
Processing Tasks Overview (#2)
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Sampling Sub-System (Mixed Analog/Digital Circuitry): Converts analog CT signal into digital DT signal ( Bandpass Sampling )
Converts real-valued RF signal into complex LP signal ( Equiv. LP Signals )Digital Front-End (Digital Processing): Convert to complex LP signal ( Equiv. LP Signals ) Equalizes response of analog front-end ( DFT-Based Filtering )
! Magnitude and Phase Bank of digital filters splits signal into sub-bands
for further processing ( Filter Banks ) Reduce sampling rate after filtering to sub-bands ( Multi-Rate Processing )
TopicsWellCover
rad/sample
DTFT of SignalBefore Digital FE
DigitalFront-End
Signals areOversampled
Processing Tasks Overview (#3)
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Detect Presence of Signal (Digital Processing): Has signal been intercepted or only noise in subband ( DFT-Based Proc. )
Estimate Parameters of Signal (Digital Processing) (see also EE522): Estimate frequency of signal ( DFT-Based Processing)
Classify/Model Signal (Digital Processing):
What type of signal is it? (Spectral Analysis of Random Signals)
je z
p p
x z a z a z a
z b z b z bb P
=
++++
++++=
L
L
2211
22
1102
1)(
PSD Model
Use PSD Model parameters { bi} and { a i} to classify signal type
Compress/De-Compress Signal (Digital Processing) (see also EE523): Exploit signal structure to allow efficient transfer( DFT-Based Proc./ Filter Banks / Spectral Analysis )
Cross-Correlate Signals (Digital Processing): Compute relative delay and Doppler ( DFT-Based Proc.& Multi-Rate Proc. )